Question

The coordinates of the vertices of the triangle are \( A(-2, 3, 6) \), \( B(-4, 4, 9) \), and \( C(0, 5, 8) \). The direction cosines of the median \( BE \) are:

• A. \( \left\langle \frac{3}{4}, 0, -\frac{2}{4} \right\rangle \)
• B. \( \left\langle -\frac{3}{\sqrt{13}}, 0, -\frac{2}{\sqrt{13}} \right\rangle \)
• C. \( \left\langle 1, 0, -\frac{2}{3} \right\rangle \)
• D. \( \left\langle \frac{3}{\sqrt{13}}, 0, -\frac{2}{\sqrt{13}} \right\rangle \)

Hint:

To find the direction cosines of a median, first calculate the midpoint of the opposite side, then use the direction ratios between the vertex and the midpoint. Normalize these ratios to obtain the direction cosines.

Answer 1

The Correct Option is D

We are given the vertices of the triangle \( A(-2, 3, 6) \), \( B(-4, 4, 9) \), and \( C(0, 5, 8) \), and we are asked to find the direction cosines of the median \( BE \), where \( E \) is the midpoint of the side \( AC \). Step 1: Find the coordinates of the midpoint \( E \) of \( AC \). The midpoint formula in three dimensions is: \[ E = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}, \frac{z_A + z_C}{2} \right) \] Substituting the coordinates of \( A(-2, 3, 6) \) and \( C(0, 5, 8) \): \[ E = \left( \frac{-2 + 0}{2}, \frac{3 + 5}{2}, \frac{6 + 8}{2} \right) = \left( -1, 4, 7 \right) \] Step 2: Find the direction ratios of the median \( BE \). The direction ratios of the median \( BE \) are the differences between the coordinates of \( B(-4, 4, 9) \) and \( E(-1, 4, 7) \): \[ \text{Direction ratios of } BE = (x_B - x_E, y_B - y_E, z_B - z_E) = (-4 - (-1), 4 - 4, 9 - 7) = (-3, 0, 2) \] Step 3: Find the direction cosines of \( BE \). The direction cosines are the normalized direction ratios. The magnitude of the direction ratios is: \[ \text{Magnitude} = \sqrt{(-3)^2 + 0^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \] Thus, the direction cosines of \( BE \) are: \[ \left\langle \frac{-3}{\sqrt{13}}, \frac{0}{\sqrt{13}}, \frac{2}{\sqrt{13}} \right\rangle = \left\langle \frac{3}{\sqrt{13}}, 0, -\frac{2}{\sqrt{13}} \right\rangle \] Therefore, the correct answer is option (D)